To analyze Talia's observed frequencies (Purple: 6, Yellow: 1, Orange: 5) and determine which statement best describes the results, let's consider each statement individually:
A. Talia's observed frequencies prove that the spinner must not be equally spaced.
- This statement implies a definitive conclusion about the spinner's fairness based on limited data. While Talia's results suggest unequal frequencies, they do not definitively prove that the spinner is not equally spaced; she would need more data for a stronger conclusion.
B. Talia's observed frequencies are equal to the expected probability of \( \frac{1}{3} \).
- This statement is inaccurate; the observed frequencies do not match the expected frequency if the spinner were equally spaced. For example, one would expect a frequency of 3 for each color if the spinner had been spun 12 times (total = 6 + 1 + 5 = 12), which does not equal the given observed frequencies.
C. If Talia increases her number of spins, the observed frequencies will likely get closer to \( \frac{1}{3} \).
- This statement is generally correct due to the Law of Large Numbers, which suggests that as the number of trials (spins) increases, the observed frequencies should converge towards the expected frequencies of \( \frac{1}{3} \) for each color.
D. If Talia increases her number of spins, the observed frequencies will likely get farther from \( \frac{1}{3} \).
- This statement is incorrect because it contradicts the Law of Large Numbers, which indicates that observed frequencies move closer to expected probabilities with more trials.
Based on this analysis, the best answer is C: "If Talia increases her number of spins, the observed frequencies will likely get closer to \( \frac{1}{3} \)."